Finite Group Databasebeta

← #215675

Subgroups — #215675

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic
2#21435C212yesmaximal
3#9893C212yesmaximal, complemented
4#9893C212yesmaximal, complemented
5#9892C212yesmaximal, complemented
6#9892C212yesmaximal, complemented
7#8721C212yesmaximal
8#8718C211yesmaximal, characteristic
9#8718C211yesmaximal, characteristic
10#2864C2212yes—
11#2864C2212yes—
12#2647C412yes—
13#2647C412yes—
14#2570—22no—
15#2570—22no—
16#2569—22no—
17#2569—22no—
18#2317C411yescharacteristic
19#2317C411yescharacteristic
20#2316C411yescharacteristic
21#2316C411yescharacteristic
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1917—22no—
28#1916—22no—
29#1767C2212yes—
30#1767C2212yes—
31#1749C2212yes—
32#1749C2212yes—
33#1745C2212yes—
34C4×#97C2212yes—
35#1295C2211yescharacteristic
36#1295C2211yescharacteristic
37#1295C2212yes—
38#1295C2212yes—
39#1295C2211yescharacteristic
40#1295C2211yescharacteristic
41#1295C2212yes—
42#1295C2212yes—
43#1295C2212yes—
44#1295C2212yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×#97—22no—
57C2×#97C2312yes—
58C2×#97C2312yes—
59C2×#97C2312yes—
60C2×#97C2312yes—
61C2×C4×C8—22no—
62#400—22no—
63#400—22no—
64#400—22no—
65#400—22no—
66#398—22no—
67#398—22no—
68#398—22no—
69#398—22no—
70#389—22no—
71#389—22no—
72#389C2×C412yes—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#385—22no—
89#385—22no—
90#384—21no—
91#384—21no—
92#384C2×C412yes—
93#382—21no—
94#382—21no—
95#382—22no—
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#382C2311yescharacteristic
99#382C2311yescharacteristic
100#382C2312yes—
101#380—22no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111#376C2×C412yes—
112C2×#95—22no—
113C43C2312yes—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128#117—42no—
129#117—42no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C411yescharacteristic
152C2×C4⋊C4C22×C411yescharacteristic
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C42—21no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—44no—
175C4⋊C8—44no—
176C22⋊C8—44no—
177C22⋊C8—44no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—44no—
187C4×C8—44no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206C24C2×C22⋊C411yescharacteristic
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C22⋊C4—24no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—21no—
272C42—21no—
273C42—22no—
274C42—22no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—42no—
282C42—42no—
283C422−1+411yescharacteristic
284C422−1+412yes—
285C422−1+411yescharacteristic
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23#41312yes—
291C23C22×C22⋊C411yescharacteristic
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—162no—
332C2×C4—162no—
333C2×C4#41312yes—
334C2×C4#51911yescharacteristic
335C2×C4#51911yescharacteristic
336C2×C4#53612yes—
337C2×C4#53612yes—
338C8—44no—
339C8—44no—
340C8—82no—
341C8—82no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—42no—
346C22—42no—
347C22#231011yescharacteristic
348C22C2×#41311yescharacteristic
349C22C2×#41311yescharacteristic
350C22#290112yes—
351C22#290112yes—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—82no—
367C4—82no—
368C2—42no—
369C2#1768611yescharacteristic
370C2#1768611yescharacteristic
371C2#1768712yes—
372C2#2183011yescharacteristic
373C2#2184612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #9893C2semidirect
4. #9893C2semidirect
5. #9892C2semidirect
6. #9892C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#353392
#214352
#98932
#98922
#87212
#87182